Problem 42
Question
Find the derivative of each function by using the Quotient Rule. Simplify your answers. $$ f(x)=\frac{x^{2}+3 x-1}{x-1} $$
Step-by-Step Solution
Verified Answer
The derivative of the function is \(\frac{x^2 - 2x - 2}{(x-1)^2}\).
1Step 1: Identify the Functions for the Quotient Rule
The Quotient Rule is used to differentiate a function that is the ratio of two functions. If you have a function \( f(x) = \frac{u(x)}{v(x)} \), the rule is:\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}\]For \( f(x) = \frac{x^2 + 3x - 1}{x - 1}\), identify:- \( u(x) = x^2 + 3x - 1 \)- \( v(x) = x - 1 \)
2Step 2: Find the Derivatives of u(x) and v(x)
Calculate the derivatives of the numerator function \( u(x) \) and the denominator function \( v(x) \). \(\frac{du}{dx} = \frac{d}{dx}(x^2 + 3x - 1) = 2x + 3\)\(\frac{dv}{dx} = \frac{d}{dx}(x - 1) = 1\)
3Step 3: Apply the Quotient Rule
Using the derivatives found in Step 2, apply the Quotient Rule:\[\frac{d}{dx}\left(\frac{x^2+3x-1}{x-1}\right) = \frac{(x-1)(2x+3) - (x^2+3x-1)(1)}{(x-1)^2}\]
4Step 4: Simplify the Expression
Expand and simplify the expression from Step 3.Calculate:\[(x-1)(2x+3) = 2x^2 + 3x - 2x - 3 = 2x^2 + x - 3\]Substituting back into the formula:\[\frac{(2x^2 + x - 3) - (x^2 + 3x - 1)}{(x-1)^2} = \frac{2x^2 + x - 3 - x^2 - 3x + 1}{(x-1)^2}\]Simplify:\[\frac{x^2 - 2x - 2}{(x-1)^2}\]
5Step 5: Finalize the Function
Ensure the expression is fully simplified. The result after simplifying is the derivative of the given function:\[\frac{x^2 - 2x - 2}{(x-1)^2}\]
Key Concepts
Understanding DerivativesSimplifying Expressions with the Quotient RuleThe Role of Applied Calculus
Understanding Derivatives
Derivatives are a fundamental concept in calculus, representing the rate of change of a function. Essentially, they tell us how a function changes with respect to its variables.
When we talk about the derivative of a function, we're looking at the slope of the tangent line to the curve of the function at any given point. The derivative gives insight into many real-world applications, such as calculating speed, acceleration, and optimization problems.
In this exercise, the focus is on using the Quotient Rule to find the derivative of a function given as a ratio of two other functions. Knowing the basic rules and properties of derivatives is crucial for understanding and simplifying complex expressions.
When we talk about the derivative of a function, we're looking at the slope of the tangent line to the curve of the function at any given point. The derivative gives insight into many real-world applications, such as calculating speed, acceleration, and optimization problems.
In this exercise, the focus is on using the Quotient Rule to find the derivative of a function given as a ratio of two other functions. Knowing the basic rules and properties of derivatives is crucial for understanding and simplifying complex expressions.
Simplifying Expressions with the Quotient Rule
The Quotient Rule is an essential tool for differentiating expressions where one function is divided by another. It is expressed as:
\[\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x) \cdot \frac{du}{dx} - u(x) \cdot \frac{dv}{dx}}{(v(x))^2}\]
To use the Quotient Rule effectively, follow these basic steps:
\[\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x) \cdot \frac{du}{dx} - u(x) \cdot \frac{dv}{dx}}{(v(x))^2}\]
To use the Quotient Rule effectively, follow these basic steps:
- Identify the numerator, \( u(x) \), and the denominator, \( v(x) \) of the function.
- Differentiate both \( u(x) \) and \( v(x) \) separately.
- Apply the Quotient Rule formula to find the derivative.
- Simplify the resulting expression by combining like terms and reducing fractions if possible.
This ensures the final expression is in its most manageable form.
The Role of Applied Calculus
Applied calculus is the practical application of calculus methods to solve problems across various fields such as physics, engineering, economics, biology, and even social sciences.
It involves using concepts of derivatives and integrals to make predictions, optimize conditions, and understand dynamic systems.
The Quotient Rule, as part of differential calculus, helps tackle problems where rate-related functions are expressed as ratios.
For instance, engineers might use it to analyze load stress in materials, while economists might determine cost-efficiency scenarios.
Mastering techniques like the Quotient Rule enables us to translate mathematical theory into actionable insights across these diverse applications. Understanding how to handle such differentiation tasks enhances problem-solving skills and guides informed decision-making in real-time applications.
It involves using concepts of derivatives and integrals to make predictions, optimize conditions, and understand dynamic systems.
The Quotient Rule, as part of differential calculus, helps tackle problems where rate-related functions are expressed as ratios.
For instance, engineers might use it to analyze load stress in materials, while economists might determine cost-efficiency scenarios.
Mastering techniques like the Quotient Rule enables us to translate mathematical theory into actionable insights across these diverse applications. Understanding how to handle such differentiation tasks enhances problem-solving skills and guides informed decision-making in real-time applications.
Other exercises in this chapter
Problem 42
For each function, find: a. \(\lim _{x \rightarrow 0^{-}} f(x)\) b. \(\lim _{x \rightarrow 0^{+}} f(x)\) c. \(\lim _{x \rightarrow 0} f(x)\) $$ f(x)=-|x| $$
View solution Problem 42
a. Find the equation of the tangent line to \(f(x)=3 x^{2}-x^{3} \quad\) at \(\quad x=1\) b. Graph the function and the tangent line on the window \([-1,3]\) by
View solution Problem 43
Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \underline{\phantom{xxx}} f(x)=x^{3}+x^{2} $$
View solution Problem 43
Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=x^{2} \sqrt{1+x^{2}} $$
View solution